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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.05029 |
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| _version_ | 1866912062111744000 |
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| author | Huynh, Dinh Tuan Linh, Tran N. K. Long, Le Ngoc |
| author_facet | Huynh, Dinh Tuan Linh, Tran N. K. Long, Le Ngoc |
| contents | We study the Waldschmidt constant of some configurations in the projective plane. In the first part, we show that the Waldschmidt constant of a set $\mathbb{X}$ of $n$ points where at least $n-3$ points among them lie on a line is either equal to $1, \frac{2n-3}{n-1}, 2, \frac{16}{7}, \frac{7}{3}, \frac{17}{7},$ or $\frac{5}{2}$. Together with the Hilbert polynomials, this gives a complete geometric characterization for $\mathbb{X}$. Next, we study some specific configurations whose Waldschmidt constants are bounded from above by $\frac{5}{2}$. Under this condition, we describe all configurations of $n$ points with $n-1$ points among them lying on an irreducible conic, and we also study some specific configurations of $9$ points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05029 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some line and conic arrangements and their Waldschmidt constants Huynh, Dinh Tuan Linh, Tran N. K. Long, Le Ngoc Combinatorics Algebraic Geometry We study the Waldschmidt constant of some configurations in the projective plane. In the first part, we show that the Waldschmidt constant of a set $\mathbb{X}$ of $n$ points where at least $n-3$ points among them lie on a line is either equal to $1, \frac{2n-3}{n-1}, 2, \frac{16}{7}, \frac{7}{3}, \frac{17}{7},$ or $\frac{5}{2}$. Together with the Hilbert polynomials, this gives a complete geometric characterization for $\mathbb{X}$. Next, we study some specific configurations whose Waldschmidt constants are bounded from above by $\frac{5}{2}$. Under this condition, we describe all configurations of $n$ points with $n-1$ points among them lying on an irreducible conic, and we also study some specific configurations of $9$ points. |
| title | Some line and conic arrangements and their Waldschmidt constants |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2410.05029 |